[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

Suppose that a straight cylindrical hole is bored from the surface of the earth through the center and out the other side. An object of mass [math]m[/math] inside the hole and at a distance [math]r[/math] from the center of the earth is attracted to the center by a gravitational force equal in absolute value to [math]\frac{mgr}{R}[/math], where [math]g[/math] is constant and [math]R[/math] is the radius of the earth. Compute the work done by this force of gravity in terms of [math]m[/math], [math]g[/math], and [math]R[/math] as the object falls

• from the surface to the center of the earth,
• from the surface of the earth through the center to a point halfway between the center and surface on the other side,
• all the way through the hole from surface to surface.

[Hint: Let the [math]x[/math]-axis be the axis of the cylinder, and the origin the center of the earth. Define the gravitational force [math]F(x)[/math] acting on the object at [math]x[/math] so that: (i) its absolute value agrees with the above prescription, and (ii) its sign agrees with the convention given at the beginning of \secref{8.5}.]